In the next preview assignment and in the next class, you will need to be able to calculate and interpret the value of the chi-square test statistic.
Calculating and Interpreting the Chi-Square Test Statistic
In the previous in-class activity, we calculated the value of the chi-square test statistic. In this corequisite support activity, we will build upon this calculation using a simplified example in order to understand the statistic’s purpose and meaning.
Question 1
1) Imagine you are playing a dice gambling game. Each die has 6 sides. If you roll a 1or 2, you win. If you roll a 3 or 4, it’s a tie. If you roll a 5 or 6, your opponent wins. You play, and you end up losing money. You want to see if the die is unfairly weighted towards the higher numbers.
Part A: You want to conduct a formal test to see whether the die isunfairly weighted. Formulate twotestable hypotheses for this scenario.
Part B: The following tables show different possible results for different numbers of total rolls. Do the results provide convincing evidence that the dieis weighted? Rank the results in order from most convincing to least convincing and explain.Hint: Pay attention to both the total number of rolls (sample size) and the imbalance of results between categories.
| Table A | You Win (1,2) | Tie (3,4) | You Lose (5,6) |
| Number of Rolls | 2 | 4 | 4 |
| Table B | You Win (1,2) | Tie (3,4) | You Lose (5,6) |
| Number of Rolls | 200 | 400 | 400 |
| Table C | You Win (1,2) | Tie (3,4) | You Lose (5,6) |
| Number of Rolls | 332 | 334 | 334 |
| Table D | You Win (1,2) | Tie (3,4) | You Lose (5,6) |
| Number of Rolls | 311 | 342 | 347 |
Question 2
2) Let’s calculate the chi-square test statistic using the counts inTables A andB.
Part A: Fill out the following tables (including the expected count and scaled squared differencecolumn) and calculate thevalue of thechi-square test statistic for Tables A andB (recallthatthe chi-square test statistic is the sum of the values in the final column). Fill out the expected counts under the assumption of a fair die.
| For Table A | Expected Count | Observed Count | |
| You Win (1, 2) | 2 | ||
| Tie (3, 4) | 4 | ||
| You Lose (5, 6) | 4 |
| For Table B | Expected Count | Observed Count | |
| You Win (1, 2) | 200 | ||
| Tie (3, 4) | 200 | ||
| You Lose (5, 6) | 400 |
Part B: Look at your calculations. Which table had the greater value for thechi-square test statistic and why did it end up being so comparatively high?
Part C: Does a greater chi-square test statisticvalueindicate more convincing or less convincing evidence of a weighted die? Explain how you can tell using your calculations from Part A.
Part D:It is sometimes said the chi-square test statistic “quantifies how much our actual results defied our expectations.” Explain how the calculations behind the statistic achieve this goal. For Table AExpected CountObserved Count(Observed−Expected)𝟐ExpectedYou Win (1, 2)2Tie (3, 4)4You Lose (5, 6)4For Table BExpected CountObserved Count(Observed−Expected)𝟐ExpectedYou Win (1, 2)200Tie (3, 4)200You Lose (5, 6)400
Question 3
3) The chi-square test statistic values are as follows (you’ll fill in the values for Tables A andB based on your answersto the previous question):
| Table A | Table B | Table C | Table D | |
| Chi-Square | 0.008 | 2.28 |
Part A: Rank the tables from highest to lowest chi-square value. Is this order similar to the order you gave in Question 1? Explain why it is similar or why it may be different.
Part B: If certain conditions are met (we will discuss these conditions in the next in-class activity), we can compare our chi-square test statistic values to the chi-square distribution to get the probability of finding theset of rolls that we observedor one that differsmore from our expectations by chance alonewhen the die is actually fair. Let’s do this for the chi-square statistic value from Table D.Goto theDCMPChi-Square Distributiontool at https://dcmathpathways.shinyapps.io/ChisqDist/.Click the Find Probabilitytab at the top of the tool. Choose the appropriate degrees of freedom(number of categories -1)and select the “Upper Tail” probability type. Enter the calculated chi-square statistic for Table D and write downthe probability displayed in the data analysis tool to the nearest whole number.
Part C: Would you say that Table D provides convincingevidence that the dieis unfairly weighted? Explain.
